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Properties

Label 315.2.j.e
Level $315$
Weight $2$
Character orbit 315.j
Analytic conductor $2.515$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(46,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.j (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + (\beta_{3} + \beta_{2} + \beta_1) q^{10} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{11} + (2 \beta_{3} - 2) q^{13} + (4 \beta_{2} + 2 \beta_1 + 3) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} - 2 \beta_1 q^{19} + (2 \beta_{3} - 1) q^{20} - 2 q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{23} + \beta_{2} q^{25} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{26} + (2 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 3) q^{28} + q^{29} - 6 \beta_{2} q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + ( - 4 \beta_{3} + 6) q^{34} + (\beta_{3} + 2 \beta_1 + 1) q^{35} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{38} + ( - 3 \beta_{2} - \beta_1 - 3) q^{40} + ( - 2 \beta_{3} + 5) q^{41} + (\beta_{3} + 5) q^{43} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{44} + \beta_{2} q^{46} + (2 \beta_{2} + 2) q^{47} + ( - 4 \beta_{3} + 5 \beta_{2} + \cdots + 5) q^{49}+ \cdots + (3 \beta_{3} + 9 \beta_{2} + 7 \beta_1 + 8) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} - 6 q^{16} + 4 q^{17} - 4 q^{20} - 8 q^{22} - 2 q^{23} - 2 q^{25} - 12 q^{26} - 22 q^{28} + 4 q^{29} + 12 q^{31} - 6 q^{32} + 24 q^{34} + 4 q^{35} + 8 q^{38} - 6 q^{40} + 20 q^{41} + 20 q^{43} - 12 q^{44} - 2 q^{46} + 4 q^{47} + 10 q^{49} - 4 q^{50} + 20 q^{52} - 8 q^{53} + 8 q^{55} - 18 q^{56} + 2 q^{58} - 8 q^{59} + 6 q^{61} + 24 q^{62} - 28 q^{64} - 4 q^{65} - 22 q^{67} + 20 q^{68} - 10 q^{70} + 16 q^{71} - 4 q^{73} + 32 q^{76} - 28 q^{77} - 24 q^{79} + 6 q^{80} + 18 q^{82} - 4 q^{83} + 8 q^{85} + 6 q^{86} - 4 q^{88} - 6 q^{89} - 28 q^{91} - 12 q^{92} - 4 q^{94} + 24 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.207107 − 0.358719i 0 0.914214 − 1.58346i 0.500000 + 0.866025i 0 −1.62132 − 2.09077i −1.58579 0 0.207107 − 0.358719i
46.2 1.20711 + 2.09077i 0 −1.91421 + 3.31552i 0.500000 + 0.866025i 0 2.62132 + 0.358719i −4.41421 0 −1.20711 + 2.09077i
226.1 −0.207107 + 0.358719i 0 0.914214 + 1.58346i 0.500000 − 0.866025i 0 −1.62132 + 2.09077i −1.58579 0 0.207107 + 0.358719i
226.2 1.20711 − 2.09077i 0 −1.91421 − 3.31552i 0.500000 − 0.866025i 0 2.62132 − 0.358719i −4.41421 0 −1.20711 − 2.09077i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.e 4
3.b odd 2 1 35.2.e.a ✓ 4
7.c even 3 1 inner 315.2.j.e 4
7.c even 3 1 2205.2.a.n 2
7.d odd 6 1 2205.2.a.q 2
12.b even 2 1 560.2.q.k 4
15.d odd 2 1 175.2.e.c 4
15.e even 4 2 175.2.k.a 8
21.c even 2 1 245.2.e.e 4
21.g even 6 1 245.2.a.g 2
21.g even 6 1 245.2.e.e 4
21.h odd 6 1 35.2.e.a ✓ 4
21.h odd 6 1 245.2.a.h 2
84.j odd 6 1 3920.2.a.bv 2
84.n even 6 1 560.2.q.k 4
84.n even 6 1 3920.2.a.bq 2
105.o odd 6 1 175.2.e.c 4
105.o odd 6 1 1225.2.a.k 2
105.p even 6 1 1225.2.a.m 2
105.w odd 12 2 1225.2.b.h 4
105.x even 12 2 175.2.k.a 8
105.x even 12 2 1225.2.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a ✓ 4 3.b odd 2 1
35.2.e.a ✓ 4 21.h odd 6 1
175.2.e.c 4 15.d odd 2 1
175.2.e.c 4 105.o odd 6 1
175.2.k.a 8 15.e even 4 2
175.2.k.a 8 105.x even 12 2
245.2.a.g 2 21.g even 6 1
245.2.a.h 2 21.h odd 6 1
245.2.e.e 4 21.c even 2 1
245.2.e.e 4 21.g even 6 1
315.2.j.e 4 1.a even 1 1 trivial
315.2.j.e 4 7.c even 3 1 inner
560.2.q.k 4 12.b even 2 1
560.2.q.k 4 84.n even 6 1
1225.2.a.k 2 105.o odd 6 1
1225.2.a.m 2 105.p even 6 1
1225.2.b.g 4 105.x even 12 2
1225.2.b.h 4 105.w odd 12 2
2205.2.a.n 2 7.c even 3 1
2205.2.a.q 2 7.d odd 6 1
3920.2.a.bq 2 84.n even 6 1
3920.2.a.bv 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 10 T + 17)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots + 14161 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots + 18496 \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 161)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
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